Ask a New Question
Menu

A dilation of  AB¯¯¯¯¯¯¯¯  occurs using a scale factor of  12  and a center of dilation at the origin. Prove that A′B′¯¯¯¯¯¯¯¯¯¯ is half the length of  AB¯¯¯¯¯¯¯¯  by finding the points  A′  and  B′ .(1 point)ResponsesA′ is at (1,2) and B′ is at (3,3).upper A prime  is at  left parenthesis 1 comma 2 right parenthesis  and  upper B prime  is at  left parenthesis 3 comma 3 right parenthesis .A′ is at (1,5) and B′ is at (3,6).upper A prime  is at  left parenthesis 1 comma 5 right parenthesis  and  upper B prime  is at  left parenthesis 3 comma 6 right parenthesis .A′ is at (−1,−2) and B′ is at (−3,−3).upper A prime  is at  left parenthesis negative 1 comma negative 2 right parenthesis  and  upper B prime  is at  left parenthesis negative 3 comma negative 3 right parenthesis .A′ is at (4,8) and B′ is at (12,12).

1 answer

To determine the points \( A' \) and \( B' \) after a dilation of the segment \( \overline{AB} \) with a scale factor of \( 12 \) centered at the origin, we need to first find points \( A \) and \( B \). The scale factor indicates that the new points will be \( 12 \) times further from the origin than the original points.

The dilation formula is given by:

\[ (x', y') = (kx, ky) \]

where \( (x, y) \) are the original coordinates, \( k \) is the scale factor, and \( (x', y') \) are the coordinates after dilation.

In this case, we have:

\[ k = 12 \]

Finding \( A' \) and \( B' \):

If we let \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \), then after dilation we have:

\[ A' = (12 x_A, 12 y_A) \] \[ B' = (12 x_B, 12 y_B) \]

To show that \( A'B' \) is half the length of \( AB \):

We need to establish that \( A'B' \) is half the length of \( AB \). We can express the lengths as follows:

  • Length \( AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \)
  • Length \( A'B' = \sqrt{(12 x_B - 12 x_A)^2 + (12 y_B - 12 y_A)^2} = 12 \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \)

However, note that you're asking to prove that \( A'B' \) is half the length of \( AB \), which is actually incorrect based on our formulae, as \( A'B' = 12 \times AB \).

Given that the options provided don't specify \( A \) and \( B \), the key to proving this would be by choosing particular values for \( A \) and \( B \). For instance, if we take:

  • Let \( A = (1, \frac{1}{6}) \) and \( B = (3, \frac{1}{4}) \): Then after dilation we would find:
  • \( A' = \left( 12 \cdot 1, 12 \cdot \frac{1}{6} \right) = (12, 2) \)
  • \( B' = \left( 12 \cdot 3, 12 \cdot \frac{1}{4} \right) = (36, 3) \)

This way, \( A'B' = 12 \times AB\) indeed, hence double-check the interpretations of the scale factor correctly here.

Conclusion

None of the choices provided correctly represent this scenario where \( A'B' \) would satisfy being half the length of \( AB \) via the scale factor as inputted. Thus, it seems there is miscommunication regarding whether we measure segment \( A'B' \) as double, or if we consider a different manipulation or point mappings that may define those test values.